Defects and patterns in the Calculus of Variations

本研究计划的目的是对物理学中出现的变分问题进行严格的数学分析，并对相关的偏微分方程组（PDE）的解进行分析。对于许多物理系统，可实现的配置是那些最小化（全局或局部）能量函数的配置。我们将研究的泛函包括：金兹堡-朗道模型，它是在超导的背景下引入的，也是凝聚态物理和材料中许多现象的范例；向列液晶的Landau-de Gennes模型；二嵌段共聚物的大田-川崎能量。这些模型具有许多基本特征，当模型中的一个或多个参数被发送到零或无穷大时，这些特征在奇异摄动极限中显示出来。在这些极限状态下，可以观察到解的几何奇点，如涡旋、偏斜或畴壁，这些缺陷给出了系统的最显著特征。

The object of this research proposal is the rigorous mathematical analysis of variational problems arising in physics, and of the solutions of the associated systems of partial differential equations (PDE). For many physical systems the realizable configurations are those which minimize (globally or locally) an energy functional. Among the functionals we will study are: Ginzburg-Landau models, introduced in the context of superconductivity, but also a paradigm for many phenomena in condensed matter physics and materials; the Landau-de Gennes model for nematic liquid crystals; and the Ohta-Kawasaki energy for di-block copolymers. These models have many fundamental features which reveal themselves in singular perturbation limits, when one or more parameters in the model are sent to zero or infinity. In these limiting regime the solutions are observed to develop geometrical singularities, such as vortices, disclinations, or domain walls, and these defects give the most salient features of the system. The overall goal is to develop new analytical tools to study singularly perturbed variational problems, to shed light on physical phenomena observed in experiments and simulations. The emphasis will be on variational problems for vector-valued functions, for which the Euler-Lagrange equations will be systems of nonlinear PDE. Many tools normally employed in studying a single PDE do not extend to systems, and so the analytical challenges are considerable but the mathematical interest is great. For instance, in the Landau-de Gennes theory of liquid crystals the states are described by Q-tensor fields (symmetric traceless matrix-valued functions,) and the singularities often occur because of eigenvalue crossing. This presents analytical hurdles in identifying and localizing singularities (line and ring defects, or "boojums" lying on boundary surfaces) which have subtle energy signatures. Further, mathematical models of block copolymers incorporate essential nonlocal interactions, which adds an extra layer of complexity to the PDEs describing these materials. Solving these problems will entail the development of new methods for systems of nonlinear and nonlocal PDE. I will blend my own ideas with innovations coming from nonlinear and geometric analysis, including sharp energy bounds (via vortex-ball constructions); monotonicity and eta-ellipticity methods (developed for harmonic maps); bifurcation techniques; Gamma-convergence techniques (giving limiting energies characterizing the shape and interactions of singularities); and concentration-compactness methods (which break down minimizers with complex structure into component pieces.) The analytical results obtained will give a more complete and reliable understanding of these models and the phenomena they describe, while providing new perspectives on the rich interplay between analysis, geometry, and physics.